9
votes
3answers
250 views

Dimensions of self-affine sets

Let $A$ be a $2\times 2$ matrix which we assume to be contracting, i.e., the exists $\alpha\in(0,1)$ such that $$ \|A {\mathbf x}\|_2\le \alpha\|{\mathbf x}\|_2,\quad \forall {\mathbf x}\in\mathbb ...
0
votes
1answer
81 views

Constructing measures with support in a given set

I've recently come across the Frostmann Lemma (http://en.wikipedia.org/wiki/Frostman_lemma). Its proof involves constructing a measure with certain properties on a given subset of $\mathbb{R}^n$ (I'm ...
9
votes
0answers
117 views

Decay rate of measures on Cantor set

I've read that Kahane and Salem show that if $\mu$ is any measure supported on the ternary Cantor set, then $\hat{\mu}(\xi) \not\to 0$ as $|\xi| \to \infty$, however I have been unable to find a ...
11
votes
4answers
706 views

Fourier decay rate of Cantor measures

For $0<\theta<\frac{1}{2}$, denote by $C_\theta$ the Cantor set with dissection ratio $\theta$, i.e. the Cantor set obtained from dissection parttern $(\theta, 1-2\theta,\theta)$. It is known ...
3
votes
5answers
840 views

Reference for the iterated function system of the Koch snowflake

Let $KS$ be the Koch snowflake. This fractal has an iterated function system (IFS) of the form $$ KS = \bigcup_{0 \leq k \leq 6} f_k(KS) $$ with $$ f_0(z)=\frac{1}{\sqrt{3}} e^{i\pi/2} z $$ and for $0 ...